one last time concerning ‘tables’ with a little question at the end…

I can’t do any more of this ‘tables’ stuff, but I am asked about the issue so much…

Look, here’s a ‘tables’ square, slightly adjusted:


I’ve removed the 1x and 10x sectors. 1x is trivial, 10x needs special treatment.

If you fully appreciate the flip rule you can forget the grey airbrushed section.

The square numbers in the blue-green squares are a special beautiful group, well worth studying. Many patterns and much al-jebr live here…

The rest are in 8 columns, from 1 to 8. Add up the numbers from 1 to 8 in your head and it’s 36. [I did 8×9 and halved it).

The orange numbers can be found by doubling from single digit numbers. (Conversely by halving from the products).

The blue by treblings from single digit numbers.

There are 4 spaces left, 5×6, 5×7, 6×7, 6×9.

double 15, the two primes, 5 and 7 make 35, double 21 and double 27.

Study George Cuisenaire’s original product wall chart:


You can see all the doublings.  Look carefully.

These products, the numbers in black, form

MILESTONES in the unknown territory to 100

By studying them as numbers, as we have previously discussed, their inner structures will become apparent, thus lessening the awful stress on memory so prevalent in today’s schools.

In addition by briefly studying the Primes to 100 it will become apparent to the little yellow belts that many numbers are rich in factors and many are not. They will generate a ‘feel’ for the rich numbers, learn their inner structures by familiarity and learn, as an aside, the so called ‘tables’ relevant to that number.

This however takes


even in the minds of  teachers, never mind the administrators


because even you and I dear reader


huatou: ‘Am I free?’

chronic boredom and irrelevance

please excuse this rant

a diatribe concerning the captive consumers of our woe, the children

They are the future. We create the future through them. They do not choose to be there. They do not choose to study the available curriculum. It has been chosen for them. They do not choose the structures of schooling including the structure of the day, with whom they are taught, their timetables nor the time for which individual subjects are taught. They do not choose just what is to be taught from the infinities of possibilities, nor why. They do not choose to divide knowledge into the separate subjects. They do not choose their teachers, the teachers’ personalities and the teachers’ particular teaching styles and the theories of learning on which they are (presumably) based. They rarely choose their own work. They are marked and assessed according to criteria they do not choose, and are categorized accordingly.  They are embedded in a system, a technological machine, which I imagine they have to tacitly assume is the best possible type of machine that society can construct, through much deep thought, which will enable them to live happy fulfilled lives and fully prepare them for the future. They accept it because it is the way things are. The status quo. The embedding in this prevailing school machine technology has inevitable consequences. Many like it. After all it is a vast social flux, they make friends, often for life, they meet people all the time. It is busy and involving. No one likes their work all the time after all, unless they are very lucky, and for much of the time, it is ‘OK’. It can lead to qualifications which are basically passports to further qualifications and restricted better jobs, even vocations and ‘professional’ activities and lifestyles. Children see this and it causes a certain degree of concentration (which is a kind of motivation imitating intrinsic motivation), even great concentration in some who can achieve excellent results in their exams. These achieve because they are born with the capacities that enable them to relatively easily fit in. It ‘suits’ them, as it ‘suited’ me up to a point. Everyone is pleased and their self-esteem becomes high. It is a good feeling. All you have to do is work hard and learn things and do well in exams, though exams are ‘not what anyone in their right mind would actually choose’. Still, they get on with it. If they make it to A levels or university, similar structures and processes prevail. If they decide to work hard and play hard, they generally have a good time. After all, they have the advantage of choosing (to some extent) their general areas of study. Sometimes, even specifics.  For others the story is not quite so rosy. They perceive, early on that the work is hard. That it is not very interesting. They look around and see others doing it easily and being rewarded in various ways. They get rewarded too, but as they get older they cannot help comparing themselves to those who will obviously ‘do better’ in the long run. As the work is ‘hard’ to understand and hence it is not easy to ‘do well’ it begins to be chronically wearing. Sooner or later most students experience this. Even for those who have inclinations and innate abilities praised and needed by the system, the relentless learning of new things, often not seeming to have much if any direct relevance to them and their needs, except for the need to pass exams, can cause chronic weariness can set in. It happens at university too. “Mum, why can it seem so irrelevant and dull? Why does it have to be done, apart from the exam angle? Why is it all considered to be necessary? Why is it all compulsory?  What is it all actually for?” Sometimes, they have ‘learning difficulties’. If they are told, or perceive that they are in fact in a different ‘category’ to most, they might easily come to ‘believe it’ hence totally altering their view of themselves, often for life. They might have some real special differences that are obvious, such as for instance, blindness, deafness or cerebral palsy which undoubtedly provide challenges to themselves and the providers of experience, their teachers. Sometimes, they have ‘learning difficulties’ that are less tangible, but which are still ‘seen’ to cause ‘difficulties’ for the ‘normal’ processes of the technological machine system. It is rarely seen as a product of the system itself. The system is generally not sensitively reflexive enough for that. It is the fault of the others – parents, society in general, the children in particular. Sometimes through individual teachers’ enthusiasms, and sometimes because the children like the teachers as people, students can get carried along down even very abstract and esoteric routes and actually enjoy the processes and products of learning along the way getting satisfaction through the sense of a job well done. There is nothing wrong with that, but it doesn’t always happen. In fact, it usually doesn’t happen. Some work is just too dull, or presented in too dull a style, in a flat, matter of fact, monotonous manner, ‘because its good for you’. It can be perceived as irrelevant by the students. Some teachers just cannot hold it all together. Other issues can emerge. ‘Disruption’ of the ‘lesson’ can easily occur due to others who have already given up the idea of pursuing the lessons’ aims and deciding to ‘have a bit of fun’ instead. It is true that it takes only one very disruptive person to destroy a teaching environment. Almost nothing can productively proceed in an atmosphere of noise and confusion. Threats and rewards can be issued and will ‘work’ with some at least temporarily. Machine gun nests work a treat. The lid can be kept on through externally imposed discipline. The repressive communist regime of Tito’s Yugoslavia managed this quite well. More ‘discipline’, including clearly visible, increasingly painful sanctions, parent/carer contracts and now quite common well established systems of ‘assertive discipline’ are acclaimed, and can ‘hold it all together’. Guilt and fear also work a treat. ‘Order’, real and apparent, is re-establishable. The normal processes of the status quo school technological machine are then re-established and proceed to the apparent satisfaction and indeed praise of all, often including the students. Certainly to most staff, the general population and government, who are seen to be improving standards. Indeed, at one level they are. Life is once again a bed of roses……but is it? Symptoms have been suppressed. What about the causes? Who bothers digging here? The causes are more insidious, more chronic, more pathological. They are potential viruses threatening the system itself. Leave them alone. In order to dig where it is dark one requires an active conscience, an unflagging, instinctual need for the development of self and professional knowledge. The digger requires courage, because what one might find in the dug hole might backfire on ones own beliefs and inner, tacitly held convictions. The digger might archaeologically expose himself and be taken away as a deviant. That is what societies do to maintain the deeply caused and little sensed status quo. It creates ‘objects of deviance’ which are cast out. Suck on these ideas and see for yourselves if there may be any substance in them. Do any cause resonances within you about teaching and learning. I am just one agent in this flux, self-constructed by my labours and experiences. Chronic boredom and irrelevance infect present educational systems, at all levels, but especially for the those compelled to be captive audiences. Dig where it is dark. Most people aren’t interested in this esoteric activity, because not only is it difficult, it is disturbing. It is also, transformative. Who wants that?

against too much prescribed content and the learning of undesirable outcomes

National curriculae have strong tendencies. A National curriculum might have as one of its aims to improve the access of all children to what is seen as a desirable amount of knowledge. That is fine, BUT…

National curriculae are mostly generated by advice from specialists tempered by ideologies of the prevailing political system. The first ensures an immense amount of material to be covered, because specialists live within their created worlds and love them so they can think of lots to do and the second ensures that a) the lists sound similar to what they did at school and b) great emphasis is given to processes, procedures and organisations for ‘learning’ that these power-possessing beings have had experience of. So, kids have a curriculum that is 1000 miles wide and an inch deep (maybe a foot) fragmented by Victorian attitudes and organisations for learning. The Victorian attitudes include the fragmentation of the world into a few subject areas, the fragmentation of the child’s time into standard units such as hours, and the fragmentation of the people of the world into ghettos they call groups or streams. The children quickly learn from these ancient environments who is smart or dumb, who is destined to be a king or a pauper, who will live to serve and who will serve, etc.



There are better ways…

current material factors inhibiting robust learning

There is, in spite of everything, powerful learning proceeding within some classrooms, but the prevailing forces of coercion and shallow curriculae inhibit it intensely and prevent renewal.

The material factors inhibiting it, to be more precise include:

A shallow, extremely wide curriculum not emphasizing robust learning in depth

Fragmented mechanical coercive organisation of the teaching day

Fragmented subject areas containing fragmented progressions of ideas

Not enough time available for in depth exploration due to inappropriate immense demands of the shallow curriculum structure

Teachers having to teach for national summative testing

The dismantling of in-service and primary teacher education services for ideological reasons

conscious and unconscious interference by government in the state education system

Comparison through too much summative testing generating feelings of superiority and inadequacy, the creation of fear due to inadequate fast ‘progress’ through too much surface content, competition rather than collaboration, visible signs of class structure through inappropriate grouping methods, fragmentation of the love of enquiry and the generation of ‘subject’ areas and endless content lists, inadequate and politicised curriculum change generated by those who do not know, inadequate systems of development and support for teachers and assistants, the generation of mass factory-market culture, the destruction of teachers’ lives through coercion and paperwork, the destruction of the state education system through shallow self- interest ideology……

All these are hallmarks of an inadequate education system.

Our education system.

the journey 1

If you engage with arithmetic, or anything else, and you really want to know, you will begin a journey. It will probably last a lifetime. Once you have become a black belt, seventh dan, you may choose once again to wear the white belt. If you are a consultant doctor, you become Mr.

One gains knowledge and wisdom but never reaches an end. There can be no end. If you think you have reached the end you are mistaken.

It is clearly the same with engaging in the Cui way.

It is, or should be, not a way that is prescribed. You have to make it your own. As such, your way will be different in many respects to others’ interpretations of the way. This is normal and proper. There is nothing unusual about it. Everyone must start somewhere. Do not compare yourself to anyone else. It is your journey.

There are so many inappropriately incomplete and partial journeys generated by the present education system.

I live in a city. Where is this city? It exists in every inhabitants mind. There are as many cities where I live as inhabitants of the city. Clearly it can be located on a map, but the knowledge and understanding of the city and all that it means is personal.

This is the difference between a prescribed content, as a list, and a full and robust ‘feel’ of the meaning. You have to work at this kind of learning. It is beyond facts. It engages your whole being. You have to create the meaning yourself, through wanting to. If you don’t want to, you won’t begin. No one can force you. At least, you can be forced in school up to a point through fear, guilt and external discipline, but this is an unnecessary crime.


So, just begin.

conditioning 2 – you, maths, space and time, the government etc

Whatever you have been exposed to in general society, whatever you have exposed yourself to through conversations, books, courses and every other transaction you have made with the world, others, the environment and your self, will have CONDITIONED you and INFLUENCED you. Whilst this is obvious to a gibbon, people in general have strong tendencies to believe that the manner in which they view and respond to the world is THE ONE TRUE PATH. Clearly this is the extreme view and your path is clearly ONE PATH, But…these paths are deeply entrenched. They have a tendency to become YOUR TRUTH.

These ‘paths’ can be considered to be ‘interpretative networks’ through which the ‘real world’ is perceived and then conceived. These interpretative networks are filters through which ‘a world’ is conceived. As such these networks ‘create’ the ‘world’ which you ‘see’. You do not see the ‘real world’ because you cannot. All you can ever see is your version.

That is not to say that if you are not sitting properly in Silent Illumination, the overseeing zen priest will not hurt you when he whacks you on the shoulder with his whacking stick. It will hurt. The stick is ‘real enough’…

The ‘brute reality’ exists at a level that will effect you. Even when discussing the nature of this ‘brute reality’ however, never mind the mental realms (where our main interest lies), the colour we perceive as red, for example, DOES NOT EXIST ‘out there’. It is a construct of our perceptual apparatus. There is no red out there. There might well be frequencies of what we call electromagnetic radiation that can be considered to be isomorphic with this ‘red’, but red itself does not exist except in our minds.

We do not see things as they are but how we are enabled to see them.

If we are considering the ‘outer world’ in general for a mo’, consider this then: that which we perceive as space and time, also like red,  DO NOT EXIST ‘out there’, they are categories of perception that come into existence when we as humans do. I.e. when we are born (or maybe in the womb). Kant figured that one out. However, even though these categories are overwhelmingly powerful, controlling and utterly ‘obvious’ to us in our general day to day life, they do not really EXIST ‘out there’. They certainly exist for us, we have cornered the market in them, but they are created categories.

How much more created then are our beliefs about MATHS?

What you have experienced throughout your school lives and everything you have been exposed to since has undoubtedly conditioned you into believing that what you understand by ‘maths’ IS maths.

IT IS OBVIOUSLY NOT THE ONLY WAY of conceiving this game that can be called mathematics.

You have to accept that even though there is a vast government approved WAY of thinking and speaking about and then of teaching about this thing called maths, it is only ONE way, not THE way.

Unfortunately, the government way is a thousand miles wide and one inch deep.

There is too much in the curriculum, and it is done too quickly and in a shallow manner, resulting in fear and confusion for too many.

It does not, except in the rarest of individuals who can get through without too much damage, lead to a creative approach to the subject.

The Cuisenaire Way is a creative way of viewing maths as something which can be created, composed and improvised with, to use a musical metaphor. One can learn to write music, not play a few restricted tunes.

The Cui Way is a light year wide and bottomless.

You will need to peer into your own conditioning and reconsider…

Unfortunately you will be beaten with sticks if you do not follow the government line, and the government line is inadequate. The government and its representatives are not in the business of being truly open, honest and willing to learn, of reconsidering their conditioning.

You however, probably are.

Good on ya sport!

Read on…